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Yeah having the heater on will cool the engine as you're drawing heat away. As the engine temp drops the thermostat should close to reduce the flow of coolant
That sounds low. Should be 12.5 - 13.5v really. Get someone to start the car whilst you measure the voltage.
As Daz says, a low battery causes some very strange faults, and from what I've seen batteries are only lasting 3-5 years ish. So if yours hasn't been replaced then I'd defo go with that...